Many Republican lawmakers, however, fear the strategy could result in a government shutdown that would hurt their party’s brand.

Johanns said linking a possible government shutdown to the quest to defund ObamaCare has boosted the national profiles of some senators, but he warned it would not yield much benefit for other Republicans.

“It burst them to a national profile. They had a lot of talkers on radio, etc., talking about it,” he said. “My feeling is we just need to be honest with people. This has zero chance of being successful.”

Rubio responded Wednesday by framing the fight over ObamaCare as the biggest issue facing the Congress.

“There’s a disagreement about whether it’s the right tactic,” he said. “If there’s one issue that we should be willing to do everything we can, it should be this one. If there’s one issue that we should be willing to take to the limit, it’s this one. It’s that bad for the country.”

Rubio held events across Florida during the August recess to highlight what he sees as the disastrous economic consequences of the law.

Tensions are rising in the Senate Republican conference over the best tactical approach to defunding the healthcare law because some members have become the targets of pressure ads from conservative groups.

Cruz urged using the government funding resolution as leverage to defund ObamaCare when he met with conservative activists at the Family Leadership Summit in Iowa last month. He’s not letting up despite his colleagues’ discomfort.

“We’re certainly continuing to press the case on every front,” he said Wednesday.

“It’s amazing that whenever an elected official takes a principled stand, the establishment questions their motives,” he said.

None of three possible 2016 White House candidates spoke in favor of the government shutdown strategy at a Senate Republican meeting Wednesday where ObamaCare was a topic, according to one lawmaker who attended.

“That wasn’t the proper forum, with McConnell being targeted by ads because of their effort,” the source said.

The effort to use a possible government shutdown as leverage has put Paul in a tricky situation. He has signed a letter advocating for the tactic but he’s also an ally of McConnell, who could face a tough challenge in his state’s primary election next year. McConnell’s challenger in that race, Matt Bevin, has ripped the minority leader on ObamaCare.

When asked about the potential political impact on McConnell, Paul said: “Everybody has their own ideas about the best strategy for how to defeat ObamaCare, so I haven’t been part of any criticism of other senators for what strategy they determine is best.”

Cruz on Tuesday slammed a proposal offered this week by House GOP leaders as ineffective because it would force the Senate to vote on defunding ObamaCare before considering an extension of government funding. The House plan would not make government funding contingent on defunding the law.

One Republican who attended Wednesday’s meeting said many of his colleagues have yet to take a position on the House resolution, which lacks the votes to pass the lower chamber. Some GOP members want to push for delaying ObamaCare while others prefer defunding it.

There is growing concern among Senate Republicans about funding levels in the House bill, which would set spending at $986.3 billion through December, according to a Senate source. Several senators say that would exceed the $967 billion spending cap set by the 2011 Budget Control Act.